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Relating definitions of {$e$}

The most basic definition of {$e$} is as the limit of {$(1+\frac{1}{n})^n$}.

For each n, the terms give a row of Pascal's triangle.

Viewing from the left, the terms have the form

{$\frac{n(n-1)\cdots (n-k)}{k!}$}

and so multiplying by {$\frac{1}{n^k}$} and taking the limit yields {$\frac{1}{k!}$}.

{$e$} is then the sum of these terms.